print barcode image c# PLOTTING THE GRAPH OF A FUNCTION in Software

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PLOTTING THE GRAPH OF A FUNCTION
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Until we learn some more sophisticated techniques, the basic method that we shall use for graphing functions is to plot points and then to connect them in a plausible manner EXAMPLE 131
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Sketch the graph of f (x) = x 3 x
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SOLUTION We complete a table of values of the function f
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x 3 2 1 0 1 2 3 y = x3 x 24 6 0 0 0 6 24
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We plot these points on a pair of axes and connect them in a reasonable way (Fig 141) Notice that the domain of f is all of R, so we extend the graph to the edges of the picture EXAMPLE 132
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Sketch the graph of f (x) =
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if x 2 if x > 2
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SOLUTION We again start with a table of values x 3 2 1 0 1 2 3 4 5 y = f (x) 1 1 1 1 1 1 3 4 5
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We plot these on a pair of axes (Fig 142) Since the de nition of the function changes at x = 2, we would be mistaken to connect these dots blindly First notice that, for x 2, the function is identically constant Its graph is a horizontal line For x > 2, the function is a line of slope 1 Now we can sketch the graph accurately (Fig 143) You Try It: Sketch the graph of h(x) = |x| 3 x
CHAPTER 1 Basics
Fig 141
EXAMPLE 133
Sketch the graph of f (x) =
x +1
SOLUTION We begin by noticing that the domain of f , that is the values of x for which the function makes sense, is {x: x 1} The square root is understood to be the positive square root Now we compute a table of values and plot some points
Basics
Fig 142
AM FL Y
Fig 143
EXAMPLE 134
Sketch the graph of x = y 2
x 1 0 1 2 3 4 5 6
x+1 0 1 2 3 2 5 6 7
Connecting the points in a plausible way gives a sketch for the graph of f (Fig 144)
CHAPTER 1 Basics
Fig 144
SOLUTION The sketch in Fig 145 is obtained by plotting points This curve is not the graph of a function
Fig 145
A curve that is the plot of an equation but which is not necessarily the graph of a function is sometimes called the locus of the equation When the curve is the graph of a function we usually emphasize this fact by writing the equation in the form y = f (x) You Try It: Sketch the locus x = y 2 + y
40 184
CHAPTER 1 COMPOSITION OF FUNCTIONS
Basics
Suppose that f and g are functions and that the domain of g contains the range of f This means that if x is in the domain of f then f (x) makes sense but also g may be applied to f (x) (Fig 146) The result of these two operations, one following the other, is called g composed with f or the composition of g with f We write (g f )(x) = g(f (x))
f (x)
g ( f (x))
Fig 146
EXAMPLE 135 Let f (x) = x 2 1 and g(x) = 3x + 4 Calculate g f SOLUTION We have (g f )(x) = g(f (x)) = g(x 2 1) ( )
Notice that we have started to work inside the parentheses: the rst step was to substitute the de nition of f , namely x 2 1, into our equation Now the de nition of g says that we take g of any argument by multiplying that argument by 3 and then adding 4 In the present case we are applying g to x 2 1 Therefore the right side of equation ( ) equals 3 (x 2 1) + 4 This easily simpli es to 3x 2 + 1 In conclusion, g f (x) = 3x 2 + 1 EXAMPLE 136 Let f (t) = (t 2 2 )/(t + 1 ) and g(t) = 2t + 1 Calculate g f and f g SOLUTION We calculate that (g f )(t) = g(f (t)) = g t2 2 t +1 ( )
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